Showing papers on "Ricci decomposition published in 2020"
TL;DR: It is shown that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Riccisoliton is trivial.
Abstract: In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of connected trivial Ricci solitons by imposing different restrictions on the energy function. We also use Hessian of the potential function to characterize compact trivial Ricci solitons. Finally, we show that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Ricci soliton is trivial.
18 citations
TL;DR: It is proved that generalized ( κ, μ) -paracontact metric manifolds satisfying certain flatness conditions on the ℳ M -projective curvature tensor are characterized by certain structures whose properties are discussed in some detail.
Abstract: Abstract We consider generalized ( κ , μ ) {(\\kappa,\\mu)} -paracontact metric manifolds satisfying certain flatness conditions on the ℳ {\\mathcal{M}} -projective curvature tensor. Specifically, we study ξ- ℳ {\\mathcal{M}} -projectively flat and ℳ {\\mathcal{M}} -projectively flat generalized ( κ , μ ) {(\\kappa,\\mu)} -paracontact metric manifolds and, further, ϕ- ℳ {\\mathcal{M}} -projectively symmetric generalized ( κ ≠ - 1 , μ ) {(\\kappa\
eq-1,\\mu)} -paracontact metric manifolds. We prove that they are characterized by certain structures whose properties are discussed in some detail.
4 citations
Posted Content•
TL;DR: In this article, the authors introduce the notion of rigidity for harmonic-Ricci solitons and provide some characterizations for rigidity, generalizing some known results for Ricci solITons, in the compact case and in the complete non-compact case.
Abstract: In this paper we introduce the notion of rigidity for harmonic-Ricci solitons and we provide some characterizations of rigidity, generalizing some known results for Ricci solitons. In the compact case we are able to deal with not necessarily gradient solitons while, in the complete non-compact case, we restrict our attention to steady and shrinking gradient solitons. We show that the rigidity can be traced back to the vanishing of certain modified curvature tensors that take into account the geometry a Riemannian manifold equipped with a smooth map $\varphi$, called $\varphi$-curvatures, which are a natural generalization of the standard curvature tensors in the setting of harmonic-Ricci solitons.
2 citations